On the recommendation of the Broadcast Television Systems Committee (BTSC), the audio signal standard adopted for Multichannel Television Sound incorporates compression/expansion of the stereophonic difference signal and the secondary audio program (SAP) signal. At the transmitter the stereophonic difference signal and the SAP signal are compressed to enhance signal to noise ratio. Receivers adapted to faithfully reproduce these signals include expander circuitry which operate on the stereophonic difference and SAP signals with a transfer function which is the inverse of the compression function. The expander transfer function, T(f,b), according to the BTSC standard is defined by the equation ##EQU1## where j is the conventional imaginary operator, f is the signal frequency and b is the root mean square value of the signal to be expanded.
The transfer function T(f,b) requires a number of division operations which, in general, require relatively complex circuitry to implement. This is particularly so if the audio signal is to be processed in digital format. It has been determined that a digital transfer function H(Z,k) which closely approximates the transfer function T(f,b) is defined by ##EQU2## where Z is the conventional Z transform variable, A is a mapping constant equal to 2.pi.(20.1 KHz)/26(f.sub.s) and k is related to the root mean square variable "b" by the equation ##EQU3## The factor f.sub.s in the mapping constant A is the sample rate of the digital audio samples.
The transfer function H(Z,k) may be realized without using any variable divider circuits, thus, simplifying the expander circuitry. However, such an expander system requires apparatus to derive the variable, k, from the root mean square value, b, of the signal to be expanded. This invention is directed toward deriving the control variable, k, from the variable b.
From the function defining the variable k it will be seen that a straight forward generation of the variable k from the variable b requires division. Division in real time processing is undesirable because of circuit complexity. It is therefore an object of this invention to derive the variable, k, with a minimum of circuitry hardware.